Structural Transitions in Dense Networks

TL;DR

This paper studies a network growth model showing a phase transition from sparse to dense networks at p=1/2, with complex structural anomalies and conditions for complete graphs as network size grows.

Contribution

It introduces a new evolving network model with probabilistic copying, revealing phase transitions and structural anomalies in network density and clique formation.

Findings

Networks are sparse for p<1/2 and dense for p≥1/2.

Structural anomalies occur at p=2/3, 3/4, 4/5, etc.

Probability of complete graphs remains non-zero as N approaches infinity.

Abstract

We introduce an evolving network model in which a new node attaches to a randomly selected target node and also to each of its neighbors with probability $p$. The resulting network is sparse for $p<\frac{1}{2}$ and dense (average degree increasing with number of nodes $N$) for $p\geq \frac{1}{2}$. In the dense regime, individual networks realizations built by this copying mechanism are disparate and not self-averaging. Further, there is an infinite sequence of structural anomalies at $p=\frac{2}{3}$, $\frac{3}{4}$, $\frac{4}{5}$, etc., where the dependences on $N$ of the number of triangles (3-cliques), 4-cliques, undergo phase transitions. When linking to second neighbors of the target can occur, the probability that the resulting graph is complete---where all nodes are connected---is non-zero as $N\to\infty$.